Two solutions to two people having the same birthday

Question

I came across the birthday problem on a forum and I found that there were two answers. My knowledge of probability is pretty limited so I can't quite understand the difference.

X and Y having the same birthday (ignoring the leap year case):

Case 1.

1-(365/365 * 364/365)

I am yet to make sense of the birthday paradox and there seem to be a lot of variations where there are 1,2,...n people in a room, at a party and so on. I won't ask for the explanation here but I have one question: Why take the complement instead of calculating the answer directly? (I read some explanations but they were too mathematical and all I got was that it was difficult to do so. What's the intuition behind it?)

Case 2.

1/365

In my understanding, for X to have been born on the same day as Y, there is only one outcome, 1 in 365 days. So the answer is 1/365. (Does this case come under the birthday paradox?)

In Case 1, people seem to influence each other's probabilities (as opposed to Case 2 where Y's birthday was sort of constant?) but there seems to be some nuance to the wordings which makes these two problems different. What am I missing here?

Answer 1

Both of these cases give the same result, it is just a different way of thinking about the problem.

Case 2 is a great way to think of it for two people. The reason people often think of this problem with complements as in case 1 is that it makes it easier to solve when there are more than two people. Take the problem with three people for example. For them to be born on different days, there are 365 options for the third person, 364 for the second, and 363 for the third. So the probability is

$$1-\frac{365}{365}*\frac{364}{365}*\frac{363}{365}$$

This can easily be extended to any number of people. Using case 2 logic it is more complicated, because there are more cases to consider. All three could share a birthday, or any pair of them share could share a birthday, or none of them could. As the number of people grows, the number of cases grows and this computation gets more complicated.

Either way to think about it is fine, but the compliment trick is a good one to know as it can often simplify problems.

Answer 2

An assignment of birthdays to all $n$ people makes $1$ outcome. That's why there are $365^n$ total possible outcomes. Of these outcomes the desirable ones are those which include at least two people receiving the same birthday. The undesirable outcomes are those in which every person receives a different birthday ; there are $\frac{365!}{(365-n)!}$ such outcomes. Therefore the desired probability is $1-\frac{365!}{365^n(365-n)!}$ computed using the multiplication principle.